What role does Stella play in the conflict of the story? What is the significance of Blanche trying to convince Stella to leave Stanley?Why is...

In A Streetcar Named Desire, the
central plot and conflict come down to one battle--the battle for Stella.  At its heart,
or at least at the heart of the plot, the play is a conflict between Blanche and Stanley
for Stella. 

That's why the scene during which Blanche
tries to convince Stella she should leave Stanley is so important, and why it's so
important that Stanley hears what's going on.  It's also the reason Stanley is so intent
on finding out the "truth" about Blanche.  At stake is his wife and his way of life. 
Stella is that which the conflict is about, to answer your question specifically. 
Stella is the goal.

Of course, Stanley wins.  The
"immigrant" Pole, though not technically an immigrant, wins out against the old,
traditional South.  And what enables Stanley to win?  Desire.  Life at its heartiest, in
a sense.  Passion.  Love.  Lust, if you want to use that word, though I, personally,
would not. 

Blanche does lie, of course, but she is also a
bit deluded, employs defense mechanisms in order to cope with reality, tries to hold on
to what she and her family once was.  Her behaviors are often not something to be
admired.  But at the same time, she is much like all of us, just trying to find a way to
survive and hang on and stay sane and be well-liked, and to find
love.